Question

Let R=\begin{pmatrix}a&3&b\\\c&2&d\\\0&5&0\end{pmatrix}: a,b,c,d ∈ {0,3,5,7,11,13,17,19}. Then the number of invertible matrices in R is

Answer 1

The absolute value of matrix R is negative 5, and the absolute value of matrix P is also negative 5. The determinant ∣R∣ can be equal to zero in the following cases:
(i) Two of the values a, b, c, d are zeros, which can be (a and b), (b and d), (d and c), or (c and a) $→$ 4 × 72 ways = 196.
(ii) Any three of the values a, b, c, d are zeros $→$ 4C3​×7=28.
(iii) All four of the values a, b, c, d are zeros $→$ 1.
(iv) All four of the values a, b, c, d are non-zero but the same number $→$ 7.
(v) When two are alike and the other two are alike (non-zero) $→$ 7C2​×2×2=84.
The number of invertible matrices = 84 – 196 – 28 – 1 – 7 – 84 = 3780.

So, the answer is 3780.